HMMT 2020 T2

Let $n$ be a fixed positive integer. An $n$-staircase is a polyomino with $\frac{n(n+1)}{2}$ cells arranged in the shape of a staircase, with arbitrary size. Here are two examples of 5 -staircases:

Prove that an $n$-staircase can be dissected into strictly smaller $n$-staircases.